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Look at the equation below. Can you figure out the value of z ? In this concept, we will learn how to use the order of operations to help us to solve equations.

6 \times 3^2 \div 2 + z + 4(7 - 3) + 2^3 \div 4 = 3^2 \times 6 + 1

Guidance

The order of operations tells us the correct order of evaluating math expressions. We always do parentheses first and then exponents . Next we do multiplication and division (from left to right) and finally addition and subtraction (from left to right) .

In order to evaluate expressions using the order of operations, we can use the problem solving steps to help.

  • First, describe what you see in the problem. What operations are there?
  • Second, identify what your job is. In these problems, your job will be to solve for the unknown.
  • Third, make a plan . In these problems, your plan should be to use the order of operations.
  • Fourth, solve .
  • Fifth, check . Substitute your answer into the equation and make sure it works.

Example A

Follow the order of operations and show each step. What is the value of the variable?

 b + 2 \times 3 \times 2^2 \div 3 = 2(5 + 6) - 2

Solution:

We can use the problem solving steps to help us with the order of operations.

& \mathbf{Describe:} && \text{The equation has parentheses, exponents, multiplication, division, subtraction and addition.}\\&&& b \ \text{is the variable.}\\\\& \mathbf{My \ Job:} && \text{Apply the order of operations rule to figure out the value of}\ b.\\\\& \mathbf{Solve:} && b + 2 \times 3 \times 2^2 \div 3 = 2(5 + 6) - 2\\&&& \mathbf{Parentheses} \qquad \qquad \quad b + 2 \times 3 \times 2^2 \div 3 = 2 \times 11 - 2\\&&& \mathbf{Exponents} \qquad \qquad \quad \ b + 2 \times 3 \times 4 \div 3 = 2 \times 11 - 2\\&&& \mathbf{Multiplication/} \qquad \quad b + 8 = 22 - 2\\&&& \mathbf{Division}\\&&& \mathbf{(left \ to \ right)}\\&&& \mathbf{Addition/} \qquad \qquad \qquad \ b + 8 = 20\\&&& \mathbf{Subtraction}\\&&& \mathbf{(left \  to \ right)} \qquad \qquad \ b=20-8\\&&& \qquad \qquad \qquad \qquad \qquad \quad b=12\\\\& \mathbf{Check:} && \text{Replace} \ b \ \text{with 12 in the equation. Check that the two expressions}\\&&&\text{(to the right and to the left of the = symbol) name the same number.}\\&&& 12 + 2 \times 3 \times 2^2 \div 3 = 2(5 + 6) - 2\\&&& 12 + 2 \times 3 \times 2^2 \div 3 = 2\times 11 - 2\\&&& 12 + 2 \times 3 \times 4 \div 3 = 2\times 11 - 2\\&&& 12 + 8 = 22 - 2\\&&& 20 = 20

Example B

Follow the order of operations and show each step. What is the value of the variable?

10^2 - 6(4 + 6) - 3^2 - 3 \times 4 - 1 = d + 50 \div 5^2

Solution:

We can use the problem solving steps to help us with the order of operations.

& \mathbf{Describe:} && \text{The equation has parentheses, exponents, multiplication, subtraction, and addition.}\\&&& d \ \text{is the variable.}\\\\& \mathbf{My \ Job:} && \text{Apply the order of operations rule to figure out the value of}\ d.\\\\& \mathbf{Solve:} && 10^2 - 6(4 + 6) - 3^2 - 3 \times 4 - 1 = d + 50 \div 5^2\\&&& \mathbf{Parentheses} \qquad \qquad \quad 10^2 - 6\times 10 - 3^2 - 3 \times 4 - 1 = d + 50 \div 5^2\\&&& \mathbf{Exponents} \qquad \qquad \quad \ 100 - 6\times 10 - 9 - 3 \times 4 - 1 = d + 50 \div 25\\&&& \mathbf{Multiplication/} \qquad \quad 100 - 60 - 9 - 12 - 1 = d + 2\\&&& \mathbf{Division}\\&&& \mathbf{(left \ to \ right)}\\&&& \mathbf{Addition/} \qquad \qquad \qquad \ 18=d+2\\&&& \mathbf{Subtraction}\\&&& \mathbf{(left \  to \ right)} \qquad \qquad \ 18 -2= d\\&&& \qquad \qquad \qquad \qquad \qquad \quad d = 16\\\\& \mathbf{Check:} && \text{Replace} \ d \ \text{with 16 in the equation. Check that the two expressions}\\&&&\text{(to the right and to the left of the = symbol) name the same number.}\\&&& 10^2 - 6(4 + 6) - 3^2 - 3 \times 4 - 1 = 16 + 50 \div 5^2\\&&& 10^2 - 6\times 10 - 3^2 - 3 \times 4 - 1 = 16 + 50 \div 5^2\\&&& 100 - 6\times 10 - 9 - 3 \times 4 - 1 = 16 + 50 \div 25\\&&& 100 - 60 - 9 - 12 - 1 = 16 + 2\\&&& 18 = 18

Example C

Follow the order of operations and show each step. What is the value of the variable?

4(9 - 5) + h + 3^2 - 2^3 + 4^1 = 3^2 \times 3 - 2 \times 3

Solution:

We can use the problem solving steps to help us with the order of operations.

& \mathbf{Describe:} && \text{The equation has parentheses, exponents, multiplication, subtraction, and addition.}\\&&& h \ \text{is the variable.}\\\\& \mathbf{My \ Job:} && \text{Apply the order of operations rule to figure out the value of}\ h.\\\\& \mathbf{Solve:} && 4(9 - 5) + h + 3^2 - 2^3 + 4^1 = 3^2 \times 3 - 2 \times 3\\&&& \mathbf{Parentheses} \qquad \qquad \quad 4 \times 4 + h + 3^2 - 2^3 + 4^1 = 3^2 \times 3 - 2 \times 3\\&&& \mathbf{Exponents} \qquad \qquad \quad \ 4 \times 4 + h + 9 - 8 + 4 = 9 \times 3 - 2 \times 3\\&&& \mathbf{Multiplication/} \qquad \quad 16 + h + 9 - 8 + 4 = 27 - 6\\&&& \mathbf{Division}\\&&& \mathbf{(left \ to \ right)}\\&&& \mathbf{Addition/} \qquad \qquad \qquad \ h + 21 = 21\\&&& \mathbf{Subtraction}\\&&& \mathbf{(left \  to \ right)} \qquad \qquad \ h = 21 - 21\\&&& \qquad \qquad \qquad \qquad \qquad \quad h = 0\\\\& \mathbf{Check:} && \text{Replace} \ h \ \text{with 0 in the equation. Check that the two expressions}\\&&&\text{(to the right and to the left of the = symbol) name the same number.}\\&&& 4(9 - 5) + 0 + 3^2 - 2^3 + 4^1 = 3^2 \times 3 - 2 \times 3\\&&& 4\times 4 + 0 + 3^2 - 2^3 + 4^1 = 3^2 \times 3 - 2 \times 3\\&&& 4\times 4 + 0 + 9 - 8 + 4 = 9 \times 3 - 2 \times 3\\&&& 16 + 0 + 9 - 8 + 4 = 27 - 6\\&&& 21 = 21

Concept Problem Revisited

6 \times 3^2 \div 2 + z + 4(7 - 3) + 2^3 \div 4 = 3^2 \times 6 + 1

We can use the problem solving steps to help us with the order of operations.

& \mathbf{Describe:} && \text{The equation has parentheses, exponents, multiplication, division, and addition.}\\&&& z \ \text{is the variable.}\\\\& \mathbf{My \ Job:} && \text{Apply the order of operations rule to figure out the value of}\ z.\\\\& \mathbf{Solve:} && 6 \times 3^2 \div 2 + z + 4(7 - 3) + 2^3 \div 4 = 3^2 \times 6 + 1\\&&& \mathbf{Parentheses} \qquad \qquad \quad 6 \times 3^2 \div 2 + z + 4 \times 4 + 2^3 \div 4 = 3^2 \times 6 + 1\\&&& \mathbf{Exponents} \qquad \qquad \quad \ 6 \times 9 \div 2 + z + 4 \times 4 + 8 \div 4 = 9 \times 6 + 1\\&&& \mathbf{Multiplication/} \qquad \quad 27 + z + 16 + 2 = 54 + 1\\&&& \mathbf{Division}\\&&& \mathbf{(left \ to \ right)}\\&&& \mathbf{Addition} \qquad \qquad \qquad \ 45 + z = 55\\&&& \mathbf{(left \  to \ right)} \qquad \qquad \ z = 55 - 45\\&&& \qquad \qquad \qquad \qquad \qquad \quad z = 10\\\\& \mathbf{Check:} && \text{Replace} \ z \ \text{with 10 in the equation. Check that the two expressions}\\&&&\text{(to the right and to the left of the = symbol) name the same number.}\\&&& 6 \times 3^2 \div 2 + 10 + 4(7 - 3) + 2^3 \div 4 = 3^2 \times 6 + 1\\&&& 6 \times 3^2 \div 2 + 10 + 4 \times 4 + 2^3 \div 4 = 3^2 \times 6 + 1\\&&& 6 \times 9 \div 2 + 10 + 4 \times 4 + 8 \div 4 = 9 \times 6 + 1\\&&& 27 + 10 + 16 + 2 = 54 + 1\\&&& 55 = 55

Vocabulary

The order of operations tells us the correct order of evaluating math expressions. We always do parentheses first and then exponents . Next we do multiplication and division (from left to right) and finally addition and subtraction (from left to right) .

Guided Practice

For each problem, follow the order of operations and show each step. What is the value of the variable?

1. 2 + 5^2 \div 5 \times 1 + 0 \times 34 = 2y - 7(4 - 3)

2. 2a + 5(9 - 8) \times 2^2 = 2^2 \times 3^2 + 2

3. 9^2 - 8^2 - 16 + 4^3 + 2^2 = e(5 + 2) - 1

Answers:

1. Here are the steps to solve:

 2 + 5^2 \div 5 \times 1 + 0 \times 34 &= 2y - 7(4 - 3)\\2 + 5^2 \div 5 \times 1 + 0 \times 34 &= 2y - 7 \times 1\\2 + 25 \div 5 \times 1 + 0 \times 34 &= 2y - 7 \times 1\\2 + 5 + 0 &= 2y - 7\\7 &= 2y - 7\\7 + 7 &= 2y\\14 &= 2y\\7 &= y

2. Here are the steps to solve:

 2a + 5(9 - 8) \times 2^2 &= 2^2 \times 3^2 + 2\\2a + 5 \times 1 \times 2^2 &= 2^2 \times 3^2 + 2\\2a + 5 \times 1 \times 4 &= 4 \times 9 + 2\\2a + 20 &= 36 + 2\\2a + 20 &= 38\\2a &= 38 - 20\\2a &= 18\\a &= 9

3. Here are the steps to solve:

 9^2 - 8^2 - 16 + 4^3 + 2^2 &= e(5 + 2) - 1\\9^2 - 8^2 - 16 + 4^3 + 2^2 &= e \times 7 - 1\\81 - 64 - 16 + 64 + 4&= 7e - 1\\69 &= 7e - 1\\69 + 1 &= 7e\\70 &= 7e\\10 &= e

Explore More

For each problem, follow the order of operations and show each step. What is the value of the variable?

  1. 2m + 3 \times 9 \div 3^3 + 4(8 - 3) = 7^2 - 3 \times 6
  2. 7^2 \div 7 \times (5^2 - 17) - (2 \times 6) = 4l - 2^2 \times 5
  3. 6+3^2 \div 3 \times 2 +1 \times 5 =3y-2(5-3)
  4. 5m+2(7-6)\times 2^3 = 2^2 \times 4^2 +7
  5. 3^2-2^2-15+3^3+2^3=f(2+1)-2

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Jan 18, 2013

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Nov 05, 2014
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